Finite Group Databasebeta

← #136000

Subgroups — #136000

Order 512.

480 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21931C211yesmaximal, characteristic
2#11425C211yesmaximal, characteristic
3#11071C211yesmaximal, characteristic
4#8862C212yesmaximal, complemented
5#7253C212yesmaximal, complemented
6#2905C2211yescharacteristic
7C2×#407—21no—
8C22×C22⋊C8—21no—
9#2280C2211yescharacteristic
10#1850—22no—
11#1850—22no—
12#1798—21no—
13#1798C2212yes—
14C2×#344—21no—
15#1742C2211yescharacteristic
16#1570—22no—
17#1458C2212yes—
18#1317C412yes—
19#1316C412yes—
20#1301—22no—
21C24×C4D411yescharacteristic
22C22×M16—21no—
23C22×M16—21no—
24C23×C8—21no—
25C23×C8—21no—
26C2×#117D411yescharacteristic
27C2×#117C2311yescharacteristic
28#431—22no—
29#424—42no—
30#423—22no—
31C2×C4⋊C8—21no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8D412yes—
34C2×C4⋊C8D412yes—
35#407—42no—
36#407—42no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—42no—
40C2×C22⋊C8—42no—
41C2×C22⋊C8—42no—
42#381C2×C411yescharacteristic
43#381Q811yescharacteristic
44#380—22no—
45#379C2×C411yescharacteristic
46#379Q811yescharacteristic
47C4×C22⋊C4—22no—
48#344—42no—
49#343—22no—
50#336—22no—
51#336—22no—
52#334—44no—
53#324—22no—
54#323C2×C414yes—
55C25C4⋊C411yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—41no—
61C23×C4—41no—
62C23×C4—41no—
63C23×C4—41no—
64C23×C4C4⋊C411yescharacteristic
65C23×C4SD1612yes—
66C23×C4SD1612yes—
67C23×C4C2×D411yescharacteristic
68C2×M16—22no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16—41no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8—41no—
86C22×C8—81no—
87C22×C8—81no—
88C22×C8—81no—
89C22×C8—81no—
90#117—41no—
91#117—41no—
92#117—41no—
93#117—41no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4#4112yes—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4C4⋊C411yescharacteristic
99C2×C22⋊C4C4⋊C411yescharacteristic
100C2×C22⋊C4C22×C411yescharacteristic
101C2×C22⋊C4C2×Q811yescharacteristic
102C2×C42—21no—
103C2×C42C2×D412yes—
104C4⋊C8—24no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—81no—
110C4⋊C8—81no—
111C22⋊C8—24no—
112C22⋊C8—44no—
113C22⋊C8—48no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116C4×C8—42no—
117#95—22no—
118#95—22no—
119C24—21no—
120C24—21no—
121C24—41no—
122C24—44no—
123C24—41no—
124C24#10612yes—
125C24C2×C4⋊C411yescharacteristic
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—81no—
149C22×C4—81no—
150C22×C4—81no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4#10612yes—
155C22×C4C4×D412yes—
156C22×C4#12011yescharacteristic
157C22×C4C22⋊Q812yes—
158C22×C4#12311yescharacteristic
159C22×C4C2×SD1611yescharacteristic
160C22×C4C2×SD1611yescharacteristic
161C22×C4AGL(1,ℤ/8)11yescharacteristic
162C22×C4#13711yescharacteristic
163M16—42no—
164M16—42no—
165M16—81no—
166M16—81no—
167M16—81no—
168M16—81no—
169M16—81no—
170M16—81no—
171C2×C8—44no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—81no—
183C2×C8—81no—
184C2×C8—84no—
185C2×C8—84no—
186C4⋊C4—42no—
187C4⋊C4—81no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—44no—
191C22⋊C4—44no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C42—22no—
195C42—22no—
196C42—41no—
197C42—41no—
198C23—24no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—42no—
203C23—42no—
204C23—44no—
205C23—42no—
206C23—44no—
207C23—41no—
208C23—41no—
209C23—44no—
210C23—81no—
211C23—84no—
212C23—82no—
213C23#38511yescharacteristic
214C23C2×#10611yescharacteristic
215C23#42811yescharacteristic
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4#44811yescharacteristic
246C2×C4#45011yescharacteristic
247C2×C4#45311yescharacteristic
248C2×C4#45511yescharacteristic
249C2×C4B(4,𝔽2)11yescharacteristic
250C2×C4#45811yescharacteristic
251C2×C4#48111yescharacteristic
252C2×C4#48311yescharacteristic
253C8—82no—
254C8—84no—
255C8—82no—
256C8—82no—
257C8—82no—
258C22—22no—
259C22—24no—
260C22—21no—
261C22—21no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22—44no—
266C22—44no—
267C22—42no—
268C22—44no—
269C22—42no—
270C22—81no—
271C22—82no—
272C22—84no—
273C22#185311yescharacteristic
274C22#185611yescharacteristic
275C22#186711yescharacteristic
276C4—21no—
277C4—21no—
278C4—42no—
279C4—42no—
280C4—42no—
281C4—81no—
282C4—81no—
283C4#159611yescharacteristic
284C4#160111yescharacteristic
285C4#160811yescharacteristic
286C4#161311yescharacteristic
287C2—21no—
288C2—21no—
289C2—44no—
290C2—41no—
291C2—41no—
292C2#885411yescharacteristic
293C2#1296212yes—
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×C22⋊C4C22×C4
Frattini, cosocleC2×#117C23
socleC22#1867
prime radicalC25C4⋊C4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #8862C2semidirect
5. #7253C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#219312
#114252
#110712
#88622
#72532